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In this paper, surface photometry and dynamical properties of Lenticular galaxies will be developed and applied to NGC3245. In this respect, we established new relation between the intensity distribution I and the semi-major axis a Moreover, some basic statistics of both independent and the dependent variables of the relation are also given. In addition to the I(a) relation , the Sérsic r
^{1/n} model is applied for the intensity profile I(r) resulting in an estimation of the effective radius, re, and the surface brightness it encloses, μe. Both relations (I(a) and I(r)) are accurate as judged by the precision criteria which are: the probable errors for the coefficients , the estimated variance of the fit and the Q value (the square distance between the exact solution and the least square estimated solution) where all very satisfactory. Correlation coefficients between some parameters of the isophotes are also computed. Finally as examples of applications of surface photometry we determined the dynamical properties: mass, density, potential distributions, as well as distributions of escape and circular speeds in terms of Sérsic model.

Surface photometry is a bidimensional broadband technique to quantitatively describe the light distribution of extended objects like galaxies and HII regions. It is a technique rather than a distinct field of research. Reynolds [

Surface photometry is extremely important since it helps us to get information on galactic colors and its implied ages and metallicity gradients [

Surface photometry is usually based on fitting ellipses to the isophotes of galaxies especially for ellipticals and lenticulars whose their isophotes show little deviation from being perfect ellipses. Several software packages and tools can perform surface photometry; among them are GALPHOT [

The package that concerns us here is the ISOPHOTE. The ISOPHOTE’s principle task is the ELLIPSE task which does the essential role of fitting the elliptical isophotes to the galaxy image. In addition to ELLIPSE, ISOPHOTE includes some “parameter set” tasks that control the process of ELLIPSE execution and other tasks that test the ELLIPSE performance by examining its results. The algorithm upon which ELLIPSE is based and how to deal with the various tasks is well described in [

On the other hand, intensity profile I(r) plays an important role in finding the distributions density, mass, and potential which play a key role in the understanding of the galactic dynamics. We will derive these dynamical properties in terms of intensity profile I(r), which is the second goal of this paper. This is done by fitting it by a suitable model. A number of models have been put to describe the relation I(r), easily obtained from I(a), the most accepted ones are the Sérsic model for bulges and the exponential law for disks.

In the present paper, we applied the surface photometry on the g-band image of the galaxy NGC 3245 obtained from the Sloan Digital Sky Survey (SDSS). The resulted data are shown in Appendix B. Its intensity profile is well fitted by the Sérsic model at n = 2.9, then , we Substituted by Sérsic formula in our derivations for dynamical properties.

NGC 3245 (UGC 5663) is a late S01 galaxy. It is composed of an extremely bright, mildly active nucleus surrounded by a lower-surface brightness (but still very bright), smooth lens ending with a diffuse, faint outer envelope. The nucleus is spherical while both the lens and the envelope have an E5-like flattened structure [20-22]. Jian Hu [

Section 2 describes the basic formulations including the linear least-square modeling of data and some basic statistics. Section 3 presents the new relation for I(a). Section 4 gives a brief description on the Sérsic model and the results of fitting. In Section 5, we derive in detail various dynamical quantities in terms of I(r). The conclusion is given in Section 6. Finally, statistical analysis of ELLIPSE output data is shown in Appendix A.

Let z be represented by the general linear model of the form where are linearly independent functions of x. Let c be the vector of the exact values of the c's coefficients and the least-squares estimators of c obtained from the solution of the normal equations of the form The coefficient matrix is symmetric positive definite ,that is , all its eigen values are positive. Let denotes the expectation of f and the variance of the fit, defined as:

where

N is the number of observations, y is the vector with elements and has elements. The transpose of a vector or a matrix is indicated by the superscript “T”. According to the least-squares criterion, it could be shown that [

1) The estimators by the method of least-squares gives the minimum of.

2) The estimators of the parameters c, obtained by the method of least-squares are unbiased; i.e.

3) The variance–covariance matrix of the unbiased estimators is given by:

4) The average squared distance between and c is:

Finally it should be noted that if the precision is measured by the probable error then

A coefficient of correlation is a statistical measure of the degree to which two variables x and y (say) are related to each other. In case of a linear equation, the coefficient of correlation is (e.g. [

where

indicates that the two variables are totally correlated, R = 0, no relationship between them, and indicates that there is a trend between the two variables x and y. The sign of R indicates whether y is increasing or decreasing when x increasing, while its magnitude indicates how well the linear approximation is.

For data analysis of the present paper we used some basic statistics of these are:

1) Descriptive statistic; 2) Location Statistics; 3) Dispersion statistics; 4) Shape statistics.

Autocorrelation is important in time series analysis. Let be the autocorrelation at lag k. An estimate of is [

where is the mean of the data.

In what follows, new relation describing the intensity profile along the semi-major axis, I(a) for the lenticular galaxy NGC3245 will be established in the sense of least–squares criterion of Section 2.1. The data used for this relation are the first two columns of

Moreover, some basic statistics of independent variable (a) and the dependent variable (I) of this relation are given in Appendix A.

The relation and its error analysis are:

.

.

.

.

The intensity distribution along the equivalent radius can be expressed by Sérsic model [29,30]

where r_{e} is the effective radius, the radius encloses half of the whole of the galaxy, I_{e} is the intensity at this radius, and b_{n} can be given by the expression

Detailed deduction and approximation of b_{n} is discussed in Graham & Driver [

Starting by the above expression of I(r), Caon et al. [

by using the formula

hence, by definition, µ_{e} is the surface brightness at r_{e}.

By fitting the intensity profile of the SDSS g-band image of NGC3245 by Sérsic model we found that the profile is well fitted at n = 2.9.

Using the first three columns of

.

From the values of c_{1} and c_{2}, we get the effective radius r_{e} = 18.95″, whereas µ_{e} = 17.5578 mag/arcsec^{2}.

The dynamical properties of galaxies can be easily obtained if the intensity profile along radius I(r) is available. As I(r) is the main output from the surface photometry technique, distributions of properties like density, mass, potential, escape and circular velocities can be found as follows:

The density distribution is given by

where γ is the mass to light ratio.

Sérsic defined I(r) as

If we defined k = b_{n}/r_{e}^{1/n }and A = e^{bn}, Sérsic equation can be written as

Then, ρ(r) can be written as

The mass enclosed by a given radius r is

.

Substituting by the formula of ρ(r)

.

Putting, and, then M(r) becomes

.

Let us consider the general integral

.

Integrating, by parts, we get.

Substituting for Q_{0}, Q_{1}, Q_{2}, we can deduce the relation

going back to the initial values of R and m, the final mass descriptive equation is resulted.

If n is a positive fraction , then we have to consider where [t] is the greatest integer (for our case n is taken as 3).

The potential in terms of ρ(r) is given by

where G is the gravitational constant. can be regarded as

where

by following similar integration steps as that done for mass, (r) equals

substituting by both expressions of and, the final equation of is

The escape speed is given by

substitution by results in

by substituting for the expression of and differentiation, we get

In this paper, surface photometry is applied on the lenticular galaxy NGC 3245 as a case study to resulting in the new relation Log I(a) = c_{1} + c_{2} a^{0.44} , we hope to generalize this relation in the future by applying a large descriptive sample of galaxies.

Since the intensity profile I(r) is well fitted by the Sérsic r^{1/n} model where n = 2.9, we derived relations for various dynamical properties in terms of Sérsic model, strengthening the usefulness of the surface photometry technique._{}

Both relations, I(a) and I(r), are accurate as judged by a given precision criteria based on linear least-squares fitting criterion. Correlation coefficients between some parameters of the isophotes are also computed.

I-Coefficients of correlation between ellipse task parameters Correlation coefficient between a & I= –0.430894 Correlation coefficient between a & e= 0.201083 Correlation coefficient between a & P_{a} = –0.240178_{}

Correlation coefficient between a & e = –0.81832 Correlation coefficient between a & P_{a }= 0.860549 Correlation coefficient between a & P_{a }= –0.662681 II-Statistics of the semi-major axis a II-1-Basic Descriptive Statistics

• The mean = 148

• The median (central value) = 148

• The variance = 7276.67 II-2-Location Statistics

• The geometric mean = 109.9175238162870485

• The harmonic mean = 47.0803

• The root mean square = 170.751 II-3-Dispersion Statistics

• The Variance of sample mean = 24.6667

• The standard error of sample mean = 4.96655

• The coefficient of variation = 0.576374

• The mean deviation = 73.7492

• The median deviation = 74

• Sample range = 294 II-4-Shape Statistics

• Skewness = 0

• The Pearson skewness 2 = 0

• The kurtosis = 1.79997

• The kurtosis excess = –1.20003 III-Statistics of the intensity I III-1-Basic descriptive statistics

• The mean = 269.584

• The median (central value) = 31.7

• The Variance = 842620 III-2-Location Statistics

• The Geometric mean = 38.8341456925116324

• The harmonic mean 13.3765

• The root mean square = 955.217 III-3-Dispersion statistics

• The variation of sample mean = 2856.34

• The standard error of sample mean = 53.4447

• The coefficient of variation = 3.40503

• The mean variation = 375.968

• The median variation 27.12

• Sample range = 8644.82 III-4-Shape statistics

• Skewness = 6.23942

• The Pearson skewness 2 = 0.777447

• The kurtosos = 46.9564

• The kurtosos excess = 43.9564 IV-Antocorrelation of the semi-major axis V-Autocorrelation of the intensity